Oscillation of Runge-Kutta Methods for a Scalar Differential Equation with One Delay

2012 ◽  
pp. 336-344
Author(s):  
Qi Wang ◽  
Jiechang Wen ◽  
Fenglian Fu
Keyword(s):  
Differential Equation ◽  
Runge Kutta ◽  
Scalar Differential Equation

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals On Runge-Kutta processes of high order

1964 ◽  
Vol 4 (2) ◽  
pp. 179-194 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
High Order ◽  
Runge Kutta ◽  
Image Position

An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x0+h, where y, f may be vectors.

scholarly journals On the integration processes of A. Huťa

1963 ◽  
Vol 3 (2) ◽  
pp. 202-206 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sixth Order ◽  
Order Accuracy ◽  
First Order ◽  
Runge Kutta ◽  
Image Position

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.

Application of Total Variation Diminishing for 1D Nozzle Problem

2013 ◽  
Vol 393 ◽  
pp. 872-877
Author(s):  
Fatimah Yusop ◽  
Bambang Basuno ◽  
Zamri Omar
Keyword(s):  
Differential Equation ◽  
Fluid Dynamics ◽  
Partial Differential Equation ◽  
Total Variation ◽  
Tvd Scheme ◽  
Total Variation Diminishing ◽  
Variation Diminishing ◽  
Computational Fluid Dynamics Cfd ◽  
Runge Kutta ◽  
Central Scheme

Computational fluid dynamics (CFD) is very widespread use every day as a tool in fluid flow analyses. In order to solve the Partial Differential Equation (PDE), there are few approach been introduced. The total variation diminishing (TVD) is a most popular scheme which is usually used in combination with other scheme. Therefore, this study develops CFD code by using Runge-Kutta which based on combination of central scheme and TVD scheme. Comparison was done through purely Runge-Kutta and after implemented TVD. The result shows that combination of Runge-Kutta and TVD approach are more stable as compared to purely Runge-Kutta approach.

Nested Runge-Kutta methods for the differential equation y?=F(x, y)

1992 ◽  
Vol 58 (3) ◽  
pp. 224-228
Author(s):  
N. S. Golovan' ◽  
N. Ya. Lyashchenko
Keyword(s):  
Differential Equation ◽  
Runge Kutta
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